6x^2-11x-31=0

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Solution for 6x^2-11x-31=0 equation:


Simplifying
6x2 + -11x + -31 = 0

Reorder the terms:
-31 + -11x + 6x2 = 0

Solving
-31 + -11x + 6x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-5.166666667 + -1.833333333x + x2 = 0

Move the constant term to the right:

Add '5.166666667' to each side of the equation.
-5.166666667 + -1.833333333x + 5.166666667 + x2 = 0 + 5.166666667

Reorder the terms:
-5.166666667 + 5.166666667 + -1.833333333x + x2 = 0 + 5.166666667

Combine like terms: -5.166666667 + 5.166666667 = 0.000000000
0.000000000 + -1.833333333x + x2 = 0 + 5.166666667
-1.833333333x + x2 = 0 + 5.166666667

Combine like terms: 0 + 5.166666667 = 5.166666667
-1.833333333x + x2 = 5.166666667

The x term is -1.833333333x.  Take half its coefficient (-0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
-1.833333333x + 0.8402777775 + x2 = 5.166666667 + 0.8402777775

Reorder the terms:
0.8402777775 + -1.833333333x + x2 = 5.166666667 + 0.8402777775

Combine like terms: 5.166666667 + 0.8402777775 = 6.0069444445
0.8402777775 + -1.833333333x + x2 = 6.0069444445

Factor a perfect square on the left side:
(x + -0.9166666665)(x + -0.9166666665) = 6.0069444445

Calculate the square root of the right side: 2.450906862

Break this problem into two subproblems by setting 
(x + -0.9166666665) equal to 2.450906862 and -2.450906862.

Subproblem 1

x + -0.9166666665 = 2.450906862 Simplifying x + -0.9166666665 = 2.450906862 Reorder the terms: -0.9166666665 + x = 2.450906862 Solving -0.9166666665 + x = 2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + x = 2.450906862 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + x = 2.450906862 + 0.9166666665 x = 2.450906862 + 0.9166666665 Combine like terms: 2.450906862 + 0.9166666665 = 3.3675735285 x = 3.3675735285 Simplifying x = 3.3675735285

Subproblem 2

x + -0.9166666665 = -2.450906862 Simplifying x + -0.9166666665 = -2.450906862 Reorder the terms: -0.9166666665 + x = -2.450906862 Solving -0.9166666665 + x = -2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + x = -2.450906862 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + x = -2.450906862 + 0.9166666665 x = -2.450906862 + 0.9166666665 Combine like terms: -2.450906862 + 0.9166666665 = -1.5342401955 x = -1.5342401955 Simplifying x = -1.5342401955

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.3675735285, -1.5342401955}

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